Voldemort

Sorry, but you don't need to go through all that.

g(x) = (5x + 10)(q(x)) + r(x)

r(x) has to be a constant, so let's set it equal to the variable, y.

g(-2) = y = -2560 - 16 + 2 = -2574 as our remainder. 

No, there is no need to use a calculator for this problem.

f(x) = (x^2 - 1)q(x) + r(x)

We know that r(x) is a linear equation in the form of ax + b. 

f(1) = -24 = a + b (The first term cancels out)

f(-1) = -10 = -a + b

Therefore, we get two system of equations.

a + b = -24

-a + b = -10

b = -17

a = -7

Hence, the remainder is -7x - 17. 

Use the sum of cubes factorization to get:

Suppose that sin x = x and cos x = y.

x^6 + y^6 = (x^2 + y^2)((x^2 + y^2)^2 - 3x^2y^2)

x^2 + y^2 = 1 as sin^2 x + cos^2 x is always one by the pythagorean identity.

By sin(2x) = 2/3, we can find that 2sin(x)cos(x) = 2/3

Leaving us with xy = 1/3 which finally gives us x^2y^2 = 1/9

Substituing all values, we result in:

(1)(2/3) = \(\boxed{\frac23}\)

We don't even need to know the value of z! Absolutely unnecessary information!

|w|^2 = w(cong(w))

Multiplying the conjugate of w in the second equation by the value of w given in the second equation. This gives us |w|^2 = 1

|w| = 1 as magnitudes are always nonnegative. 

Builderboi has a great way to solve it but sometimes you don't have much of those formulas memorized.

Use polynomial construction.

a + b + c = 1

ab + ac + bc = -1

abc = -1

P(x) = x^3 - x^2 - x + 1

a, b, c are the roots of this polynomial. 

a^3 = a^2 + a - 1

b^3 = b^2 + b - 1

c^3 = c^2 + c - 1

a^3 + b^3 + c^3 = (a^2 + b^2 + c^2) + (a + b + c) - 3

(a + b + c)^2 - 2(ab + ac + bc) = a^2 + b^2 +  c^2 = 3

a^3 + b^3 + c^3 = 1.

We can use the fact that for any complex number a, (a)(cong(a)) = |a|^2.

\(|5z + w|^2 = (5z + w)(5\overline{z} + \overline{w}) = 121 \)

\(25|z|^2 + |w|^2 + 5(\overline{z}w + \overline{w}z) = 121\)

\(5(\overline{z}w + \overline{w}z) = 47\)

\(|3z + 4w|^2 = (3z + 4w)(3\overline{z} + 4\overline{w}) \)

\(=793 + 12(\overline{w}z + \overline{z}w)\)

\(=793 + 564/5 \)

Take the square root of that value and you will the answer.

x^2 + y^2 + z^2 + 2(xy + yz + zx) = (x + y + z) ^2

= 21 = (x + y + z)^2

x + y + z = sqrt(21)

(Make sure you know why it's positive sqrt(21)!)

Sorry, I thought it was a multiplication sign! Woops!

1 + 3 + .... + (2n - 1) = n^2

2n - 1 = 199

2n = 200

n = 100

10000 * 2(1 + 2 + 3....  + 101) 

= 10000 * 2(5151) 

= 10000 * 10302 

= 103020000

Hint: Draw a perpendicular line from point C to segment DA. This creates a 30-60-90 triangle.